The one-dimensional Exactly 1
cellular automaton:
replication, periodicity, and
chaos from finite seeds
by Janko Gravner and David Griffeath
In the Exactly 1 cellular automaton (also known as Rule 22
), every site of the one-dimensional lattice is either in state 0 or in state 1,
and a synchronous update rule dictates that a site is in state 1 next time if
and only if it sees a single 1 in its three-site neighborhood at the current
time. We analyze this rule started from finite seeds, i.e., those initial
configurations that have only finitely many 1’s. Three qualitatively different
types of evolution are observed: replication, periodicity, and chaos. We focus
on rigorous results, assisted by algorithmic searches, for the first two
behaviors. In particular, we explain why replication is observed so frequently
and present a method for collecting the smallest periodic seeds. Some empirical
observations about chaotic seeds are also presented.
The full article (< 1 MB pdf)
Links
to Cellular Automaton Simulation Software:
Supporting
Experiment and Data Files:
Experiments
corresponding to the examples of the article, an archive of Mcell (.mcl) and Golly
(.rle) initializations
Five periodic seeds
not detailed in the article, an archive of Mcell (.mcl) and Golly
(.rle) initializations
Raw data generated
for the article, an archive of .txt and .doc files
Partially supported by the National Science Foundation and the Republic of
Slovenia’s Ministry of Science